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                <ol class="chapter"><li class="chapter-item expanded "><a href="../../preface.html"><strong aria-hidden="true">1.</strong> Preface</a></li><li class="chapter-item expanded "><a href="../../ch01/ch01.html"><strong aria-hidden="true">2.</strong> Chapter 1</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../ch01/1.1/1.1.html"><strong aria-hidden="true">2.1.</strong> Section 1.1</a></li><li class="chapter-item expanded "><a href="../../ch01/1.3/1.3.html"><strong aria-hidden="true">2.2.</strong> Section 1.3</a></li><li class="chapter-item expanded "><a href="../../ch01/1.6/1.6.html"><strong aria-hidden="true">2.3.</strong> Section 1.6</a></li></ol></li><li class="chapter-item expanded "><a href="../../ch02/ch02.html"><strong aria-hidden="true">3.</strong> Chapter 2</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../ch02/key-point/key-point.html"><strong aria-hidden="true">3.1.</strong> KeyPoints</a></li><li class="chapter-item expanded "><a href="../../ch02/2.2/2.2.html"><strong aria-hidden="true">3.2.</strong> Section 2.2</a></li><li class="chapter-item expanded "><a href="../../ch02/2.3/2.3.html"><strong aria-hidden="true">3.3.</strong> Section 2.3</a></li><li class="chapter-item expanded "><a href="../../ch02/2.4/2.4.html"><strong aria-hidden="true">3.4.</strong> Section 2.4</a></li><li class="chapter-item expanded "><a href="../../ch02/2.6/2.6.html"><strong aria-hidden="true">3.5.</strong> Section 2.6</a></li><li class="chapter-item expanded "><a href="../../ch02/2.8/2.8.html"><strong aria-hidden="true">3.6.</strong> Section 2.8</a></li></ol></li><li class="chapter-item expanded "><a href="../../ch03/ch03.html"><strong aria-hidden="true">4.</strong> Chapter 3</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../ch03/key-point/key-point.html"><strong aria-hidden="true">4.1.</strong> KeyPoints</a></li><li class="chapter-item expanded "><a href="../../ch03/3.1/3.1.html"><strong aria-hidden="true">4.2.</strong> Section 3.1</a></li><li class="chapter-item expanded "><a href="../../ch03/3.3/3.3.html"><strong aria-hidden="true">4.3.</strong> Section 3.3</a></li><li class="chapter-item expanded "><a href="../../ch03/3.4/3.4.html"><strong aria-hidden="true">4.4.</strong> Section 3.4</a></li><li class="chapter-item expanded "><a href="../../ch03/3.5/3.5.html"><strong aria-hidden="true">4.5.</strong> Section 3.5</a></li><li class="chapter-item expanded "><a href="../../ch03/3.6/3.6.html"><strong aria-hidden="true">4.6.</strong> Section 3.6</a></li><li class="chapter-item expanded "><a href="../../ch03/3.7/3.7.html"><strong aria-hidden="true">4.7.</strong> Section 3.7</a></li><li class="chapter-item expanded "><a href="../../ch03/3.8/3.8.html"><strong aria-hidden="true">4.8.</strong> Section 3.8</a></li><li class="chapter-item expanded "><a href="../../ch03/3.9/3.9.html"><strong aria-hidden="true">4.9.</strong> Section 3.9</a></li></ol></li><li class="chapter-item expanded "><a href="../../ch04/ch04.html"><strong aria-hidden="true">5.</strong> Chapter 4</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../ch04/key-point/key-point.html"><strong aria-hidden="true">5.1.</strong> KeyPoints</a></li><li class="chapter-item expanded "><a href="../../ch04/4.2/4.2.html"><strong aria-hidden="true">5.2.</strong> Section 4.2</a></li><li class="chapter-item expanded "><a href="../../ch04/4.3/4.3.html"><strong aria-hidden="true">5.3.</strong> Section 4.3</a></li><li class="chapter-item expanded "><a href="../../ch04/4.4/4.4.html"><strong aria-hidden="true">5.4.</strong> Section 4.4</a></li><li class="chapter-item expanded "><a href="../../ch04/4.5/4.5.html"><strong aria-hidden="true">5.5.</strong> Section 4.5</a></li><li class="chapter-item expanded "><a href="../../ch04/4.6/4.6.html"><strong aria-hidden="true">5.6.</strong> Section 4.6</a></li><li class="chapter-item expanded "><a href="../../ch04/4.7/4.7.html"><strong aria-hidden="true">5.7.</strong> Section 4.7</a></li></ol></li><li class="chapter-item expanded "><a href="../../ch05/ch05.html"><strong aria-hidden="true">6.</strong> Chapter 5</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../ch05/5.1/5.1.html"><strong aria-hidden="true">6.1.</strong> Section 5.1</a></li><li class="chapter-item expanded "><a href="../../ch05/5.2/5.2.html"><strong aria-hidden="true">6.2.</strong> Section 5.2</a></li><li class="chapter-item expanded "><a href="../../ch05/5.3/5.3.html"><strong aria-hidden="true">6.3.</strong> Section 5.3</a></li><li class="chapter-item expanded "><a href="../../ch05/5.4/5.4.html"><strong aria-hidden="true">6.4.</strong> Section 5.4</a></li><li class="chapter-item expanded "><a href="../../ch05/5.5/5.5.html"><strong aria-hidden="true">6.5.</strong> Section 5.5</a></li></ol></li><li class="chapter-item expanded "><a href="../../ch06/ch06.html"><strong aria-hidden="true">7.</strong> Chapter 6</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../ch06/6.1/6.1.html"><strong aria-hidden="true">7.1.</strong> Section 6.1</a></li><li class="chapter-item expanded "><a href="../../ch06/6.2/6.2.html"><strong aria-hidden="true">7.2.</strong> Section 6.2</a></li><li class="chapter-item expanded "><a href="../../ch06/6.3/6.3.html"><strong aria-hidden="true">7.3.</strong> Section 6.3</a></li><li class="chapter-item expanded "><a href="../../ch06/6.4/6.4.html"><strong aria-hidden="true">7.4.</strong> Section 6.4</a></li><li class="chapter-item expanded "><a href="../../ch06/6.5/6.5.html"><strong aria-hidden="true">7.5.</strong> Section 6.5</a></li><li class="chapter-item expanded "><a href="../../ch06/6.6/6.6.html"><strong aria-hidden="true">7.6.</strong> Section 6.6</a></li><li class="chapter-item expanded "><a href="../../ch06/6.7/6.7.html"><strong aria-hidden="true">7.7.</strong> Section 6.7</a></li></ol></li><li class="chapter-item expanded "><a href="../../ch07/ch07.html"><strong aria-hidden="true">8.</strong> Chapter 7</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../ch07/7.2/7.2.html"><strong aria-hidden="true">8.1.</strong> Section 7.2</a></li><li class="chapter-item expanded "><a href="../../ch07/7.3/7.3.html"><strong aria-hidden="true">8.2.</strong> Section 7.3</a></li><li class="chapter-item expanded "><a href="../../ch07/7.3/7.4.html"><strong aria-hidden="true">8.3.</strong> Section 7.4</a></li><li class="chapter-item expanded "><a href="../../ch07/7.5/7.5.html"><strong aria-hidden="true">8.4.</strong> Section 7.5</a></li><li class="chapter-item expanded "><a href="../../ch07/7.6/7.6.html"><strong aria-hidden="true">8.5.</strong> Section 7.6</a></li><li class="chapter-item expanded "><a href="../../ch07/7.7/7.7.html"><strong aria-hidden="true">8.6.</strong> Section 7.7</a></li></ol></li><li class="chapter-item expanded "><a href="../../ch08/ch08.html"><strong aria-hidden="true">9.</strong> Chapter 8</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../ch08/8.2/8.2.html"><strong aria-hidden="true">9.1.</strong> Section 8.2</a></li><li class="chapter-item expanded "><a href="../../ch08/8.3/8.3.html"><strong aria-hidden="true">9.2.</strong> Section 8.3</a></li><li class="chapter-item expanded "><a href="../../ch08/8.4/8.4.html" class="active"><strong aria-hidden="true">9.3.</strong> Section 8.4</a></li><li class="chapter-item expanded "><a href="../../ch08/8.5/8.5.html"><strong aria-hidden="true">9.4.</strong> Section 8.5</a></li></ol></li></ol>
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                        <h1 id="exercises-for-section-84"><a class="header" href="#exercises-for-section-84">Exercises for Section 8.4</a></h1>
<h3 id="841"><a class="header" href="#841">8.4.1</a></h3>
<p>Figure 8.10 is a simple matrix-multiplication program.</p>
<ol>
<li>Translate the program into three-address statements of the type we have been using in this section. Assume the matrix entries are numbers that require 8 bytes, and that matrices are stored in row-major order.</li>
<li>Construct the flow graph for your code from (a).</li>
<li>Identify the loops in your flow graph from (b).</li>
</ol>
<pre><code>for (i=O; i&lt;n; i++)
    for (j=O; j&lt;n; j++)
        c[i][j] = 0.0;
for (i=O; i&lt;n; i++)
    for (j=O; j&lt;n; j++)
        for (k=O; k&lt;n; k++)
            ￼c[i][j] = c[i][j] + a[i][k]*b[k][j];
</code></pre>
<p>Figure 8.10: A matrix-multiplication algorithm</p>
<h4 id="answer"><a class="header" href="#answer">Answer</a></h4>
<ol>
<li>
<p>three-address statements</p>
<pre><code>B1       1)  i = 0

B2       2)  if i &gt;= n goto(13)

B3       3)  j = 0

B4       4)  if j &gt;= n goto(11)

B5       5） t1 = n * i
         6)  t2 = t1 + j
         7)  t3 = t2 * 8
         8)  c[t3] = 0.0
         9)  j = j + 1
        10)  goto(4)

B6      11)  i = i + 1
        12)  goto(2)

B7      13)  i = 0

B8      14)  if i &gt;= n goto(40)

B9      15)  j = 0

B10     16)  if j &gt;= n goto(38)

B11     17)  k = 0

B12     18)  if k &gt;= n goto(36)

B13     19)  t4 = n * i
        20)  t5 = t4 + j
        21)  t6 = t5 * 8
        22)  t7 = c[t6]
        23)  t8 = n * i
        24)  t9 = t8 + k
        25)  t10 = t9 * 8
        26)  t11 = a[t10]
        27)  t12 = n * k
        28)  t13 = t12 + j
        29)  t14 = t13 * 8
        30)  t15 = b[t14]
        31)  t16 = t11 * t15
        32)  t17 = t7 + t16
        33)  c[t6] = t17
        34)  k = k + 1
        35)  goto(18)

B14     36)  j = j + 1
        37)  goto(16)

B15     38)  i = i + 1
        39)  goto(14)
</code></pre>
</li>
<li>
<p>flow graph</p>
<p><img src="https://f.cloud.github.com/assets/340282/1368058/d28ce9f8-397c-11e3-90ae-155b6f5f21fa.gif" alt="8 4 1-2" /></p>
</li>
<li>
<p>loops</p>
<ul>
<li>{B2, B3, B4, B6}</li>
<li>{B4, B5}</li>
<li>{B8, B9, B10, B15}</li>
<li>{B10, B11, B12, B14}</li>
<li>{B12, B13}</li>
</ul>
</li>
</ol>
<h3 id="842"><a class="header" href="#842">8.4.2</a></h3>
<p>Figure 8.11 is code to count the number of primes from 2 to n, using the sieve method on a suitably large array a. That is, a[i] is TRUE at the end only if there is no prime i^0.5 or less that evenly divides i. We initialize all a[i] to TRUE and then set a[j] to FALSE if we find a divisor of j.</p>
<ol>
<li>Translate the program into three-address statements of the type we have been using in this section. Assume integers require 4 bytes.</li>
<li>Construct the flow graph for your code from (a).</li>
<li>Identify the loops in your flow graph from (b).</li>
</ol>
<pre><code>for (i=2; i&lt;=n; i++)
    a[i] = TRUE;
count = 0;
s = sqrt(n);
for (i=2; i&lt;=s; i++)
if (a[i]) 1* i has been found to be a prime *1 {
    count++ ;
    for (j=2*i; j&lt;=n; j = j+i)
        a[j] = FALSE; 1* no multiple of i is a prime *1
    }
</code></pre>
<p>Figure 8.11: Code to sieve for primes</p>
<h4 id="answer-1"><a class="header" href="#answer-1">Answer</a></h4>
<ol>
<li>
<p>three-address statements</p>
<pre><code>B1       1)  i = 2

B2       2)  if i &gt; n goto(7)

B3       3)  t1 = i * 4
         4)  a[t1] = TRUE
         5)  i = i + 1
         6)  goto(2)

B4       7)  count = 0
         8)  s = sqrt(n)
         9)  i = 2

B5      10)  if i &gt; s goto(22)

B6      11)  t2 = i * 4
        12)  ifFalse a[t2] goto(20)

B7      13)  count = count + 1
        14)  j = 2 * i

B8      15)  if j &gt; n goto(20)

B9      16)  t3 = j * 4
        17)  a[t3] = FALSE
        18） j = j + i
        19)  goto(15)

B10     20)  i = i + 1
        21)  goto(10)
</code></pre>
</li>
<li>
<p>flow graph</p>
<p><img src="https://f.cloud.github.com/assets/340282/1368108/b06992fe-3981-11e3-9f51-24f223030901.gif" alt="8 4 2-2" /></p>
</li>
<li>
<p>loops</p>
<ul>
<li>{B2, B3}</li>
<li>{B5, B6, B10}</li>
<li>{B5, B6, B7, B8, B10}</li>
<li>{B8, B9}</li>
</ul>
</li>
</ol>
<hr />
<h3 id="note"><a class="header" href="#note">Note</a></h3>
<h4 id="1-a-demo-for-algorithm-87-determining-the-liveness-and-next-use-information-foreach-statement-in-a-basic-block"><a class="header" href="#1-a-demo-for-algorithm-87-determining-the-liveness-and-next-use-information-foreach-statement-in-a-basic-block">1. A demo for algorithm 8.7: Determining the liveness and next-use information foreach statement in a basic block.</a></h4>
<pre><code>init:

three-address statements                 symbol table

                                         symbol  live   nextuse
    i)  a = b + c                       [a,      true,  null]
    j)  t = a + b                       [b,      true,  null]
                                        [c,      true,  null]
                                        [t,      true,  null]

step1:
Attach to statement j the information currently found in the symbol table

                                         symbol  live   nextuse
    i)  a = b + c                       [a,      true,  null]
    j)  t = a + b  [t, true, null]      [b,      true,  null]
                   [a, true, null]      [c,      true,  null]
                   [b, true, null]      [t,      true,  null]

step2:
In the symbol table, set x.live = false and
                         x.nextuse = null

                                         symbol  live   nextuse
    i)  a = b + c                       [a,      true,  null]
    j)  t = a + b  [t, true, null]      [b,      true,  null]
                   [a, true, null]      [c,      true,  null]
                   [b, true, null]      [t,      false, null]

step3:
In the symbol table, set a.live = true, b.live = true and
                         a.nextuse = j, b.nextuse = j

                                         symbol  live   nextuse
    i)  a = b + c                       [a,      true,  j   ]
    j)  t = a + b  [t, true, null]      [b,      true,  j   ]
                   [a, true, null]      [c,      true,  null]
                   [b, true, null]      [t,      false, null]

step4:


                                         symbol  live   nextuse
    i)  a = b + c  [a, true, j   ]      [a,      true,  j   ]
                   [b, true, j   ]      [b,      true,  j   ]
                   [c, true, null]      [c,      true,  null]
                                        [t,      false, null]
    j)  t = a + b  [t, true, null]
                   [a, true, null]
                   [b, true, null]

step5:

                                         symbol  live   nextuse
    i)  a = b + c  [a, true, j   ]      [a,      false, null]
                   [b, true, j   ]      [b,      true,  j   ]
                   [c, true, null]      [c,      true,  null]
                                        [t,      false, null]
    j)  t = a + b  [t, true, null]
                   [a, true, null]
                   [b, true, null]

step6:

                                         symbol  live   nextuse
    i)  a = b + c  [a, true, j   ]      [a,      false, null]
                   [b, true, j   ]      [b,      true,  i   ]
                   [c, true, null]      [c,      true,  i   ]
                                        [t,      false, null]
    j)  t = a + b  [t, true, null]
                   [a, true, null]
                   [b, true, null]

</code></pre>
<h4 id="2-three-ways-to-generate-code-for-fori--0-i--n--i-statement"><a class="header" href="#2-three-ways-to-generate-code-for-fori--0-i--n--i-statement">2. Three ways to generate code for &quot;for(i = 0; i &lt; n ; i++)&quot; statement</a></h4>
<pre><code> 1) i = 0
 2) if i &gt;= n goto(9)
 3)
    ...
 7) i = i + 1
 8) if i &lt; n goto(3)
 9)


 1) i = 0
 2) goto(8)
 3)
    ...
 7) i = i + 1
 8) if i &lt; n goto(3)
 9)


 1) i= 0
 2) if i &gt;= n goto(9)
    ...
 7) i = i + 1
 8) goto(2)
 9)
</code></pre>
<p>更多可参考 RednaxelaFX 的 <a href="http://rednaxelafx.iteye.com/blog/1961217">对C语义的for循环的基本代码生成模式</a></p>

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